let k be Element of NAT ; :: thesis: for d being non zero Element of NAT
for G being Grating of d st k <= d holds
for A being Subset of (REAL d) holds
( A in cells k,G iff ex l, r being Element of REAL d st
( A = cell l,r & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
let d be non zero Element of NAT ; :: thesis: for G being Grating of d st k <= d holds
for A being Subset of (REAL d) holds
( A in cells k,G iff ex l, r being Element of REAL d st
( A = cell l,r & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
let G be Grating of d; :: thesis: ( k <= d implies for A being Subset of (REAL d) holds
( A in cells k,G iff ex l, r being Element of REAL d st
( A = cell l,r & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) ) )
assume k <= d ; :: thesis: for A being Subset of (REAL d) holds
( A in cells k,G iff ex l, r being Element of REAL d st
( A = cell l,r & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
then cells k,G = { (cell l,r) where l, r is Element of REAL d : ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) } by Def8;
hence for A being Subset of (REAL d) holds
( A in cells k,G iff ex l, r being Element of REAL d st
( A = cell l,r & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) ) ; :: thesis: verum